A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.

Thursday, August 23, 2007

Had a little doorway chat with Karl (longtime Math Lab student employee, now graduated) this afternoon about the universality (or lack of it) of mathematics: to what extent is math just waiting around for us to discover it, and to what extent is math itself an artifact of human invention, the residue that's left by the human mind as it makes its imprint on all that it encompasses? The whole conversation started when I was showing him a book of logarithm tables I picked up at a garage sale or flea market somewhere a long time ago, and I wavered indecisively between the words "invented" and "discovered" when searching for the right word to describe the initial human engagement with logarithms.

"Since you said 'invented' first," said Karl, "I can tell which camp you're in." This led to a discussion of whether mathematics can truly be universal, a position neither of us defends. Karl mentioned recent research (see this link for more info) into the language of a certain Amazonian people suggesting limits to traditional Chomskian analysis, and I let him know about Anthony F. Aveni's Uncommon sense: understanding nature's truths across time and culture (University Press of Colorado, Boulder, 2006), an interesting book I worked my way through this summer. Aveni discusses the scientific undertakings of the members of various ancient and modern societies and provides accounts of culture-specific scientific knowledge that might seem patently alien to practitioners of science as defined by the Western European Enlightenment tradition. I'll definitely be looking through that text again when I start to put together my thoughts on the history of math technology course I hope to run.

Rewind several hours: as I walked into campus this morning I thought about our discussion on the topic of "Good Proof/Bad Proof" in 280 yesterday. "Damn," I thought, "that was a nice conversation." I really felt that we got right at the meat of the matter (or whatever vegetarian substitute one would like to put in its stead), and the students themselves were quick to point out, unprompted, what it is that makes a given proof a weak one or a strong one: does it use notation correctly? Consistently? Does it prove the claimed statement in full generality? Does it use correct grammar and punctuation, use complete sentences? Does it "lead the reader" conversationally through the thought processes of the prover? All of these questions get at the issues of clarity, correctness, completeness, and cohesion, my "Four Cs" of assessing the quality of a proof. Above all else, the exercise helped them develop (oh, that meaning-laden term!) "ownership" of the process of mathematical discovery: they have the same right that I do to question the validity of a proof, to test the hypotheses of a theorem. Math's truth does not inhere in a single individual no matter how much experience that individual possesses, and even the greenest of mathematical parvenus, equipped with the right tools and techniques, may approach, with healthy skepticism, a given mathematical statement with the confidence of a professor emeritus. I think that yesterday's exercise helped folks see that, and I hope that it gave them the confidence they'll require to feel free to explore the problems we'll face the rest of the semester.